Could Dark Matter Be Fully or Partially Explained by Gravitational Waves? This is something I have wondered for a long time and cannot see why it is not a possible solution. Basically, any motion of matter (Mass may be a more accurate description here) through space time will create gravitational waves (more specifically a change in the gravitational warp of space time) that propagates out from the motion at approximately the speed of light. We are able to observe this with Black hole and Neutron Star mergers, but this should technically occur for any movement of matter through space time, however small the fluctuation may be it is there from you waving your arm, the Earth Orbiting the Sun, the sun orbiting the galactic center, etc..
My main thought is that the cumulative sum of the warping from all gravitational waves within a galaxy may sum to a portion of or all of the space/time warping we attribute to Dark Matter.
While these waves would eventually travel past the edge of the galaxy there would be a relatively stable generation/concentration of them within the galaxy at any given time. I believe this effect is only really popping up around galaxies because that is where the energy density of these waves is the most concentrated (closest to their origination point). The total energy of the gravitational wave would spread out as the wave propagates and it's energy density would then decrease (a ripple on the surface of a pond has the highest energy concentration just after it is created, once that ripple spreads out it's energy is more spread out as well). A high density of gravitational waves interacting within a galaxy should in theory look similar to some mysterious mass bending the spacetime if all we can see is the bent spacetime itself. In the ripple analogy, imagine you had may boats driving around in a lake, creating many wakes. If we looked at the course of a buoy through the lake while the lake was completely calm vs its course with many boats moving around and creating lots of wake, it would change. This is my main analogy to our dark matter measurements. We see the impact they have on the path of light in/around these galaxies which would be the impact of the boat wake on the buoy path.
This brings me to the further continuation of this theory. That this may also expand some portion of Dark energy. While on a relatively small scale the impact of these waves may appear as space warping due to mass being present, I began to think about what this would look like on a larger scale past a galaxy. Think back to the lake example from the previous section. Lets assume this was an infinite lake with small sections where tons of wake is being generated. Up close near that source would be rather chaotic, but from further away the system would have a relatively consistent output of net wake. If you measured the distance along the surface of the water from your location (from a far distance away) to the source, any wake in between would increase that distance as you have to travel along the curvature of the wake. This could be caused by ANY wake, not just the wake of the source you are looking at. There could be wake coming from a source behind you that once it passes you but not your point of interest it would still be increasing that length. This scales such that the increase in length is proportional to the distance you are from any given source of wake, the further away something is the more space exists for additional wake from all directions to lengthen the distance, that lengthened distance then allows even more wake to interact and so on. As I see it this is very similar to how dark energy works. The expansion in length between you and any given point is relative to the amount of space between you. Assuming all sources of wake (net galactic gravitational wave output) is homogeneous throughout the universe (which I believe the consensus is that the universe is homogeneous on very large scales), the expansion due to this would be constant everywhere and proportional to distance.
I would be more than happy to clarify any confusion or answer any additional questions about this theory. I will admit it is rather convenient that I think the two most mysterious things in our universe could be explained by one phenomena on different scales. I'm probably wrong about one, or the other, or BOTH. But I don't know enough of the nitty gritty physics to rule this out. As far as I know my assumptions about the way gravitational waves would look on these scales is correct, I really do think the surface of a body of water is a good analogy in this regard. I'm excited to hear what people have to say and am totally open to criticism, good or bad.
 A: There are indeed some theories trying to explain dark matter with gravitational waves, but since they were just observed directly in 2016, we still lack the experimental observation to verify them. The only basis we have yet is established mathematics and theoretical physics.
Gravity itself is a source of gravity, which is why the Einstein field equations are non-linear and which arises from the fact that gravitational waves carry energy. In comparison, electromagnetic waves don't carry charge and therefore aren't their own source, but they carry energy as well and therefore contribute to gravity.
The idea of a gravitational or electromagnetic wave held together in a bounded region of spacetime by its own field energy, called a "geon", was first studied by Wheeler in 1955, but yet only contained the idea. One can imagine the intuition behind the mathematical description, which was partly given by Brill and Hartle in 1964, as being loosely analogous to some also still unproven theories trying to explain ball lightning with a standing electromagnetic wave, which is a solution to the Maxwell equations. This was first proposed in 1955 as well, but I don't know if there was any connection. Of course, finding such a solution for the Einstein field equations is way harder.
It took until 1997 for a mathematical proof, that geon solutions of the vacuum equations do indeed exist, but have no closed expression. It is still an open question, whether such geon solutions are stable. Wheeler in 1955 already proposed the idea that geons on the quantum scale, basically gravitons under their own self-interaction, called "graviballs", could serve as a model for elementary particles. Since geons have mass, but no charge, they can indeed serve as a candidate for dark matter, if they are stable which was just studied recently in 2020.

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*J. A. Wheeler, Geons (1955), found here.


*Dieter R. Brill and James B. Hartle, Method of the Self-Consistent Field in General Relativity and its Application to the Gravitational Geon, found here.


*G. P. Perry, F. I. Cooperstock, Stability of Gravitational and Electromagnetic Geons (1999), found here.


*B. Guiot, A. Borquez, A. Deur, K. Werner, Graviballs and Dark
Matter (2020) found here.
A: Gravitational waves add to the overall energy density of the universe, just like any other form of radiation.
Generally there is expected to be a stochastic background of gravitational waves consisting both of contributions of all binary motion in the universe and gravitational waves generated during inflation. Experiments (LIGO, Virgo, pulsar timing arrays, and effect on the cosmic microwave background) have been looking for these backgrounds. Thus far they have not made any (conclusive) detection. This puts a limit on how much energy can be contained in this background. A review from earlier this year limits the fraction of the critical energy density that can consist of gravitational waves $\Omega_{GW}$ as
$$  \Omega_{GW} < 1.2\times 10^{-6} \left( \frac{H_0}{100 \mathrm{km}\, \mathrm{s}^{-1} \mathrm{Mpc}^{-1}}\right)^{-2},$$
where $H_0$ is the Hubble constant in units of $\mathrm{km}\, \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$. Compared to the fraction of expected dark matter ($\sim 0.3$) this is insignificant.
